Some recent development tendencies of mathematical modeling and numerical simulation are considered. They are stimulated as improvement of mathematical models and numerical calculation method, as unprecedented growth of computer technique power. It is shown, that investigation of numerical algorithm properties is expediently to made with using special test examples, which have analytical solutions.
In the paper for reaction-diffusion equation with Caratheodory nonlinear term under conditions, which do not guarantee uniqueness of Cauchy problem solution, we prove the global resolvability in the class of nonnegative integrable functions.
In this paper we study a classical Dirichlet optimal control problem for a nonlinear elliptic equation with the coefficients which we adopt as controls in L°°(Ω). The problems of this type have no solutions in general, so we make a special assumption on the coefficients of the state equation and introduce the class of so-called solenoidal controls. We study the stability of the above optimal control problem with respect to the domain perturbation. With this aim we introduce the concept of the Mosco-stability for such problems and study the variational properties of Mosco-stable problems with respect to different types of domain perturbations.
The solubility of the class of nonlinear optimization problems arising in image registration is discussed. The necessary optimility conditions (Euler-Lagrange equation) for such kind of problems is a nonlinear Neumann boundary value problem which is not known to have a solution in general. However, in the image registration context some assumptions can be made that let us move a little bit further in this question.
The paper deal with the solution of minimal energy optimal control problem for a parabolic equation with non-local boundary condition and a cost functional with special form. The solution to this problem is presented in the form of the series with respect to the biorthogonal Riesz basis.
In this paper, we study vector optimization problems in partially ordered Banach spaces. We suppose that an objective mapping possesses a weakened property of lower semicontinuity and make no assumptions on the interior of the ordering cone. We derive the sufficient conditions for existence of efficient solutions of the above problems and discuss the role of the topological properties of the objective space. Our main goal deals with the scalarization of vector optimization problems when the objective functions are vector-valued mappings with a weakened property of lower semicontinuity. We also prove the existence of the so-called generalized efficient solutions via the scalarization process. All principal notions and assertions are illustrated by numerous examples.
The Neumann boundary value problem for the telegraph equation in a semi-bounded domain is considered. Using the method of integral representation and the reflection method, we give the explicit description for the solution of this problem.
In this paper we study the optimal control problem associated to a linear degenerate elliptic equation with mixed boundary conditions. We adopt a weight coefficient in the main part of elliptic operator as control in BV(Ω). Since the equations of this type can exhibit the Lavrentieff phenomenon and non-uniqueness of weak solutions, we show that this optimal control problem is regular. Using the direct method in the Calculus of variations, we discuss the solvability of the above optimal control problems in the class of weak admissible solutions.
The scheme of the lower semi-continuous regularization of mappings in normed spaces is proposed. We make no assumptions on the interior of the ordering cone.
A task on the intensive deformed state(IDS) of a viscoelastic declivous cylinder, which is grown under the action of inner pressure, is considered. The process of continuous increase takes a place on an internal radius so, that a radius and pressure change on set to the given law. The special case of linear law of creeping is considered, and also numeral results are presented as the graphs of temporal dependence of tensions and moving for different points of cylinder.
We study the algorithm for the construction of mathematical descriptions of real processes, which are characterized by the system of ordinary differential equations.
We studied traffic flow models in vector-valued optimization statement where the flow is controlled at the nodes of network. We considered the case when an objective mapping possesses a weakened property of upper semicontinuity and made no assumptions on the interior of the ordering cone. The sufficient conditions for the existence of efficient controls of the traffic problems are derived. The existence of efficient solutions of vector optimization problem for traffic flow on network are also proved.
The first initial boundary-value problem for telegraph equation in bounded domain is considered. The exact solution this problem is obtained. The construction of solution is based on combination prolongation and reflection methods with integral representation vast class telegraph equation solutions which was developed earlier.
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